Can't Remove/Detatch Bones/Modifier/Vertex Group
You can either do it with a brush in Weight Paint mode, or in the Properties panel > Object Data > Vertex Groups, select the vertex group in the list, select the vertex you want to remove in the 3D view, click on Remove button under the list.

Note that on the right of the vertex groups list, under the + and -, you have a down arrow that gives you additional functions, like Remove From All Groups.